I've sent three e-mails today.
I've been sending three e-mails today.
How do I see the matter?
There's no place for "today" in the first sentence.
There's no place for both "today" and "three" in the second one.
I've sent three e-mails. (It doesn't matter when, how. I've sent them! That's all.)
I've been sending e-mails. (It doesn't matter how many, when? recently, lately, you've probably interrupted me with your question about what have I been doing, it also doesn't matter have I finished or not...)
"today" is an adverb of time and match for example simple past but not present perfect or present perfect continuous.
If "today" takes place in the expression simple past should be used with.
I sent three e-mails today.
Did ye?
Who pay for?